Semiconical sets, semihomogeneous functions, and a new duality scheme in convex analysis (Q5933390)

A set \(C\) in a locally convex space \(E\) is semiconical if \(C\) equals its semiconical hull \(\text{scon }C:= \{\lambda x: x\in C, \lambda\geq 1\}\), strictly if the closed convex hull of \(\text{scon }C\) does not contain \(0\). The negative polar of a set \(M\) is \(E^{\#}:= \{x'\in E^*:\langle x',x\rangle\leq-1\), for all \(x\in C\}\). A nonnegative real function is semihomogeneous if its epigraph is semiconical. A duality theory is set up, based on these ideas, analogous to the usual duality using convex sets and functions, and polars. A characterization is given for nonnegative functions majorized by a strictly semihomogeneous function.